FINS5513 Chap.5 Single-Index Model and Regression
Single-Index Model and Regression
Week 5 turns the CAPM relation into an empirical regression. The single-index model generally uses excess returns, with asset excess return equal to alpha plus beta times market excess return plus a residual. On the security characteristic line, alpha is the intercept, beta is the slope and each residual is the vertical deviation from the fitted line.
The model decomposes total variance into beta-squared market variance and residual variance when the residual is uncorrelated with the market. R-squared is the systematic share of total variance under that decomposition. If residuals across assets are uncorrelated, cross-asset covariance reduces to the product of their betas and market variance. This assumption is what reduces the large Markowitz covariance-input burden.
CAPM testing uses a first-pass time-series regression to estimate betas and a second-pass cross-sectional regression to test whether average excess returns are priced by beta. Results depend on coefficient uncertainty, sample choice, the market proxy and the possibility that omitted factors create alpha or residual correlation.
What this chapter covers
- 01
Excess-return single-index regression and raw-return alternative
- 02
Security characteristic line intercept, slope and residual
- 03
Alpha as benchmark-relative abnormal return or model misspecification
- 04
Total variance split into systematic and residual variance
- 05
R-squared as the systematic share of total variance
- 06
SIM covariance from asset betas and market variance
- 07
Reduced input burden relative to a full covariance matrix
- 08
First-pass beta estimates and second-pass CAPM pricing tests
Worked example · free
Alpha and a complete SIM risk decomposition
- +1Alpha is 8% minus 1.10 times 6%, which equals 1.40%.
- +1Systematic variance is 1.10 squared times 0.15 squared, which equals 0.027225.
- +1Residual variance is 0.25 squared minus 0.027225, which equals 0.035275. Residual standard deviation is 18.78%.
- +1R-squared is 0.027225 divided by 0.0625, which equals 43.56%.
Key terms
- Security Characteristic Line
- The fitted relation between asset and market returns in the single-index regression.
- Regression Alpha
- The fitted intercept measuring benchmark-relative abnormal return under the model.
- Regression Beta
- The fitted slope measuring sensitivity to the market return proxy.
- Residual Risk
- Return variation left unexplained by the market factor.
- R-Squared
- The proportion of total return variance explained by the fitted market relation.
- Market Proxy
- A broad observed index used in place of the unobservable true market portfolio.
Single-Index Model and Regression FAQ
Why does the model use excess returns?
Subtracting the risk-free rate aligns the regression with CAPM's risk-premium relation. Raw-return notation can be used, but the intercept then has a different numerical interpretation and must be identified.
How is residual risk calculated?
Compute total variance and systematic variance in squared units, subtract systematic from total variance, and take the square root only if the question requests residual standard deviation.
What does R-squared measure here?
It measures the share of asset variance associated with the fitted market component under the model. It does not measure the probability that CAPM is true or prove alpha is significant.
Why does SIM reduce covariance inputs?
When residuals are mutually uncorrelated, covariance between two assets depends only on their betas and market variance. That replaces many pairwise estimates with one common-factor structure.
What are the two CAPM test passes?
The first pass estimates each asset's beta through time. The second pass regresses average excess returns across assets on those estimated betas to test the predicted intercept and beta price.
Exam move
Start every problem by declaring whether r means raw return or R means excess return. On an SCL sketch, label alpha at the intercept, beta as the slope and residuals vertically. For numerical decomposition, square both standard deviations, calculate beta-squared market variance, subtract to obtain residual variance and reconcile the pieces to total variance.
Calculate R-squared from systematic variance divided by total variance. For covariance, state the zero residual-covariance assumption before using the beta-product shortcut. In regression interpretation, separate coefficient magnitude, statistical significance and goodness of fit. For two-pass tests, explain how first-pass estimation error and the chosen market proxy enter the second-pass verdict.
Close with the sample and model conditions rather than treating historical alpha as guaranteed future skill.
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